📚 Year 12 Edexcel Maths: Key Terminology Quick Memorisation Guide | 爱德思12年级数学：关键术语速记指南
Mastering the vocabulary of Year 12 Edexcel Mathematics is essential for understanding problem statements, writing clear solutions, and communicating mathematical ideas accurately. This guide pairs each key term with a concise English definition and its Chinese counterpart, organised by topic to help you internalise the language of pure maths, statistics, and mechanics efficiently.
掌握爱德思12年级数学的词汇术语是理解题意、写出规范解答和准确表达数学思想的关键。本速记指南按主题整理了核心术语，为每个术语提供简明的英文定义和中文解释，帮助你高效内化纯数学、统计和力学中的语言表达。
1. Algebraic Expressions | 代数表达式
Polynomial – an expression consisting of variables and coefficients involving only addition, subtraction, multiplication, and non‑negative integer exponents of variables.
多项式 – 仅包含变量的加法、减法、乘法以及非负整数次幂的表达式。
Coefficient – the numerical factor multiplying a variable or a power of a variable in a term.
系数 – 项中与变量或变量的幂相乘的数值因子。
Discriminant – for a quadratic ax² + bx + c, the discriminant is b² − 4ac; it determines the nature and number of real roots.
判别式 – 对于二次式 ax² + bx + c，判别式为 b² − 4ac；它决定了实根的性质和个数。
Completing the square – rewriting a quadratic expression in the form a(x + p)² + q to find its turning point or solve the equation.
配方法 – 将二次式改写为 a(x + p)² + q 的形式，用以求顶点或解方程。
Function – a relation where each input (x) is associated with exactly one output (y).
函数 – 每个输入值 (x) 都唯一对应一个输出值 (y) 的关系。
Domain – the set of all possible input values (x‑values) for which a function is defined.
定义域 – 函数所有可能输入值（x 值）的集合。
Range – the set of all possible output values (y‑values) a function can produce.
值域 – 函数所有可能输出值（y 值）的集合。
Inverse function – a function f⁻¹ that reverses the effect of f, satisfying f(f⁻¹(x)) = x and f⁻¹(f(x)) = x.
反函数 – 逆转原函数 f 作用的函数 f⁻¹，满足 f(f⁻¹(x)) = x 且 f⁻¹(f(x)) = x。
Composite function – a function created by applying one function to the output of another, written as fg(x) or f(g(x)).
复合函数 – 将一个函数作用于另一个函数输出值所构成的函数，记作 fg(x) 或 f(g(x))。
2. Coordinate Geometry | 坐标几何
Gradient – the measure of steepness of a line, calculated as change in y divided by change in x.
斜率（梯度） – 直线倾斜程度的量度，等于 y 的变化量除以 x 的变化量。
y‑intercept – the point where a graph crosses the y‑axis, typically the constant term in y = mx + c.
y 轴截距 – 图像与 y 轴相交的点，在 y = mx + c 中通常是常数项 c。
Equation of a straight line – commonly expressed as y = mx + c, where m is the gradient and c is the y‑intercept.
直线方程 – 通常表示为 y = mx + c，其中 m 为斜率，c 为 y 轴截距。
Perpendicular lines – two lines are perpendicular if the product of their gradients is −1 (m₁ × m₂ = −1).
垂直线 – 若两条直线斜率之积为 −1 (m₁ × m₂ = −1)，则它们互相垂直。
Circle equation – a circle with centre (a, b) and radius r has the equation (x − a)² + (y − b)² = r².
圆的方程 – 圆心为 (a, b)、半径为 r 的圆的方程为 (x − a)² + (y − b)² = r²。
Tangent – a straight line that touches a curve or circle at a single point without crossing it.
切线 – 与曲线或圆仅在一个点接触且不穿过的直线。
Normal – a line perpendicular to the tangent at the point of contact.
法线 – 在切点处与切线垂直的直线。
3. Trigonometry | 三角学
Radian – a unit of angle measurement defined as the angle subtended by an arc equal in length to the radius; π rad = 180°.
弧度 – 角度的度量单位，弧长等于半径时所对应的圆心角；π 弧度 = 180°。
Sine (sin), Cosine (cos), Tangent (tan) – fundamental trigonometric ratios defined on a right‑angled triangle or the unit circle.
正弦 (sin), 余弦 (cos), 正切 (tan) – 在直角三角形或单位圆上定义的基本三角比。
Secant (sec), Cosecant (cosec), Cotangent (cot) – reciprocal trigonometric functions: sec θ = 1/cos θ, cosec θ = 1/sin θ, cot θ = 1/tan θ.
正割 (sec), 余割 (cosec), 余切 (cot) – 三角函数的倒数：sec θ = 1/cos θ, cosec θ = 1/sin θ, cot θ = 1/tan θ。
CAST diagram – a mnemonic diagram showing which trigonometric functions are positive in each quadrant (C‑A‑S‑T).
CAST 图 – 表示各象限中三角函数值为正的记忆图示（C‑A‑S‑T 分别对应不同象限）。
Trigonometric identity – an equation true for all values of the variable, e.g. sin²θ + cos²θ ≡ 1.
三角恒等式 – 对所有变量值都成立的等式，如 sin²θ + cos²θ ≡ 1。
sin²θ + cos²θ ≡ 1
Exact values – known sine, cosine and tangent values for standard angles such as 30°, 45°, 60° (or π/6, π/4, π/3) derived from special triangles.
精确值 – 通过特殊三角形得到的标准角度（如 30°、45°、60° 或 π/6、π/4、π/3）的正弦、余弦和正切值。
4. Exponentials and Logarithms | 指数与对数
Exponential function – a function of the form f(x) = aˣ where a > 0; often the natural exponential eˣ, where e ≈ 2.718.
指数函数 – 形如 f(x) = aˣ (a > 0) 的函数；常指自然指数函数 eˣ，其中 e ≈ 2.718。
Natural logarithm (ln x) – the logarithm to base e; ln x is the inverse of eˣ.
自然对数 (ln x) – 以 e 为底的对数；ln x 是 eˣ 的反函数。
Laws of logarithms – logₐ(xy) = logₐx + logₐy; logₐ(x/y) = logₐx − logₐy; logₐ(xⁿ) = n logₐx.
对数运算法则 – logₐ(xy) = logₐx + logₐy；logₐ(x/y) = logₐx − logₐy；logₐ(xⁿ) = n logₐx。
Exponential growth and decay – models of the form A = A₀e^(kt) (growth for k > 0, decay for k < 0).
指数增长与衰减 – 形如 A = A₀e^(kt) 的模型（k > 0 为增长，k < 0 为衰减）。
Logarithmic equation – an equation in which the unknown appears inside a logarithm; solved using the laws and the definition of logs.
对数方程 – 未知数出现在对数中的方程；通过运算法则和对数的定义求解。
5. Differentiation | 微分
Derivative – the function f'(x) or dy/dx that gives the instantaneous rate of change of a function with respect to x.
导数 – 函数 f'(x) 或 dy/dx，表示函数关于 x 的瞬时变化率。
Power rule – for f(x) = xⁿ, the derivative is f'(x) = nxⁿ⁻¹.
幂法则 – 若 f(x) = xⁿ，则 f'(x) = nxⁿ⁻¹。
Chain rule – used to differentiate composite functions: if y = f(u) and u = g(x), then dy/dx = dy/du × du/dx.
链式法则 – 用于复合函数求导：若 y = f(u) 且 u = g(x)，则 dy/dx = dy/du × du/dx。
Product rule – for y = u v, the derivative is dy/dx = u dv/dx + v du/dx.
乘法法则 – 若 y = u v，则 dy/dx = u dv/dx + v du/dx。
Quotient rule – for y = u / v, the derivative is dy/dx = (v du/dx − u dv/dx) / v².
除法法则 – 若 y = u / v，则 dy/dx = (v du/dx − u dv/dx) / v²。
Stationary point – a point on a curve where the derivative is zero (dy/dx = 0); it can be a local maximum, local minimum, or point of inflection.
驻点 – 曲线上导数为零 (dy/dx = 0) 的点；可能是局部极大值、局部极小值或拐点。
Second derivative – the derivative of the derivative, denoted f”(x) or d²y/dx²; used to determine concavity and classify stationary points.
二阶导数 – 导数的导数，记作 f”(x) 或 d²y/dx²；用于判断凹凸性和区分驻点类型。
Point of inflection – a point where the curve changes concavity; may or may not be a stationary point.
拐点 – 曲线凹凸性发生改变的点；可能是也可能不是驻点。
6. Integration | 积分
Indefinite integral – the antiderivative of a function, written ∫ f(x) dx; it includes a constant of integration + C.
不定积分 – 函数的原函数，记作 ∫ f(x) dx；结果中包含积分常数 + C。
Constant of integration – an arbitrary constant added to an indefinite integral to represent the family of antiderivatives.
积分常数 – 加到不定积分中的任意常数，用以表示一族原函数。
Power rule for integration – ∫ xⁿ dx = xⁿ⁺¹/(n+1) + C, for n ≠ −1.
幂积分法则 – ∫ xⁿ dx = xⁿ⁺¹/(n+1) + C，其中 n ≠ −1。
Definite integral – an integral with upper and lower limits, ∫ₐᵇ f(x) dx, representing the signed area between the curve and the x‑axis.
定积分 – 带有上限和下限的积分，∫ₐᵇ f(x) dx，表示曲线与 x 轴之间的带符号面积。
Fundamental theorem of calculus – connects differentiation and integration: ∫ₐᵇ f(x) dx = F(b) − F(a) where F'(x) = f(x).
微积分基本定理 – 将微分与积分联系起来：若 F'(x) = f(x)，则 ∫ₐᵇ f(x) dx = F(b) − F(a)。
Area under a curve – found by evaluating a definite integral; if the curve is below the x‑axis the integral gives a negative value, so absolute area must be considered.
曲线下方面积 – 通过计算定积分求得；若曲线在 x 轴下方，积分值为负，须考虑绝对值面积。
7. Sequences and Series | 序列与级数
Arithmetic sequence – a sequence where the difference between consecutive terms is constant, called the common difference d.
等差数列 – 相邻两项之差为常数的数列，该常数称为公差 d。
Geometric sequence – a sequence where the ratio between consecutive terms is constant, called the common ratio r.
等比数列 – 相邻两项之比为常数的数列，该常数称为公比 r。
Sum to infinity – for a convergent geometric series with |r| < 1, S∞ = a / (1 − r), where a is the first term.
无穷和 – 对于收敛的等比级数，当 |r| < 1 时，S∞ = a / (1 − r)，其中 a 为首项。
Binomial expansion – expresses (a + b)ⁿ as a sum of terms using binomial coefficients; for positive integer n, (1 + x)ⁿ = Σ ⁿCᵣ xʳ.
二项展开式 – 利用二项式系数将 (a + b)ⁿ 展开为各项之和；对正整数 n，(1 + x)ⁿ = Σ ⁿCᵣ xʳ。
Factorial (n!) – the product of all positive integers up to n: n! = n × (n−1) × … × 1.
阶乘 (n!) – 小于等于 n 的所有正整数的乘积：n! = n × (n−1) × … × 1。
Binomial coefficient ⁿCᵣ – the number of ways to choose r items from n, given by n! / [r!(n−r)!].
二项式系数 ⁿCᵣ – 从 n 个中选取 r 个的组合数，计算公式为 n! / [r!(n−r)!]。
8. Vectors | 向量
Scalar – a quantity with magnitude only (e.g. mass, speed, distance).
标量 – 只有大小没有方向的量（如质量、速率、路程）。
Vector – a quantity with both magnitude and direction (e.g. displacement, velocity, force).
向量 – 既有大小又有方向的量（如位移、速度、力）。
Magnitude – the length or size of a vector, denoted |v|; for v = (x, y), |v| = √(x² + y²).
模（大小） – 向量的长度，记作 |v|；若 v = (x, y)，则 |v| = √(x² + y²)。
Unit vector – a vector with magnitude 1; often used to specify direction.
单位向量 – 模长为 1 的向量；常用于指示方向。
Position vector – a vector from the origin to a point, describing its location.
位置向量 – 从原点指向某点的向量，用于描述点的位置。
Component form – expressing a vector in terms of i and j (unit vectors in the x and y directions) as ai + bj.
分量形式 – 用 i 和 j（x 和 y 方向的单位向量）表示向量，形如 ai + bj。
Dot product (scalar product) – for vectors a and b, a·b = |a||b| cos θ; used to find angles and test perpendicularity.
点积（数量积） – 对向量 a 和 b，a·b = |a||b| cos θ；用于求夹角和判断垂直性。
9. Statistics: Data and Sampling | 统计：数据与抽样
Population – the entire set of individuals or items under investigation.
总体 – 研究中所有个体或项目的全集。
Sample – a subset of a population selected for observation.
样本 – 从总体中选出的用于观测的子集。
Random sample – a sample chosen in such a way that every member of the population has an equal chance of selection.
随机样本 – 总体中每个成员均有相等被选概率的样本。
Discrete data – data that can only take specific, countable values (e.g. number of students).
离散数据 – 只能取特定可数值的数据（如学生人数）。
Continuous data – data that can take any value within a given range (e.g. height, time).
连续数据 – 在给定范围内可取任意值的数据（如身高、时间）。
Histogram – a graphical display using bars where the area of each bar represents frequency; frequency density = frequency / class width.
直方图 – 用条块面积表示频率的图形；频数密度 = 频数 / 组距。
Box plot – a diagram showing the minimum, lower quartile, median, upper quartile and maximum of a dataset.
箱线图 – 展示数据最小值、下四分位数、中位数、上四分位数和最大值的图形。
Interquartile range (IQR) – the difference between the upper and lower quartiles; a measure of spread.
四分位距 (IQR) – 上四分位数与下四分位数之差，一种离散程度的度量。
10. Probability and Discrete Random Variables | 概率与离散随机变量
Random variable – a variable whose value is a numerical outcome of a random phenomenon; usually denoted by a capital letter like X.
随机变量 – 其值为随机现象数值结果的变量，通常用大写字母如 X 表示。
Probability distribution – a table, function or graph that gives the probabilities of all possible values of a discrete random variable.
概率分布 – 以表格、函数或图形形式
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